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Some useful formulae in screw theory for robotics

  • rodrigues: Given a vector, w, and an angle, theta, computes the exponential map of an angle using Rodrigues' formula. This is akin to the rigid body transformation of a body about an axis w, through a point q, by an angle theta.

  • skewsem: Computes the antisymmetric or skew symmetric matrix from a 3D vector w = [wx, wy, wz].

  • twist: Given an infinitesimal linear generator v and an infinitesimal angular generator w, computes the twist vector \in R^6 as [(-w x v) w]

  • twistmap: Given a twist, \zeta, and an angle of rotation \theta, computes the exponential map of the twist exp(\hat{zeta}, theta), where $\hat{\zeta}$ is the isomorphism of the twist linear algebra in R^6

  • curvature_form: Computes the relative curvature at the point of contact of two objects. This is typically useful in multifingered kinematics or IAB manipulations such as I used in my PhD thesis. This implementation is from Spivak's differential geometry book, 1979.

  • psi_dot: computes the relative orientation between the tangent planes of two coordinate charts in R^2. The rotation angle through an angle \psi aligns the manipulating object's x-axis to the x-axis of the manipulated object. \psi is so chosen such that a rotation about the z-axis by \psi aligns the xy-axes of both objects (see Montana, 1988).

  • contact_kinematics: A simple illustration of the contact dynamics between two rigid, or semi rigid objects in R^3. The examples assume that the objects are spheres with single coordinate charts as defined in [Murray, 1990].

  • bvp_free: comptes the contact free boundary value deformation for a desired extension/compression of the SoRO model Result is the pressure

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Of screws and rigid bodies and differential kinematic curves

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